308 ANALYTICAL MECHANICS
path with a period equal to . The axes of the path coin-
cide with the coordinate axes.
Case IV. When d = ± - and 6 = a; the path becomes a cir-
cle, and the motion uniform circular motion with a period
1 4. 27T
equal to -
Find the resultant motion due to the superposition of the motions defined by the following equations:
(1) x a cos o}t and y = a sin at.
(2) x a cos (t + tQ) and y = a sin ~~ t.
(3) re = a sin coi and T/ = a cos (£ + tQ).
(4) a; = a sin (art + 5) and y = a cos (art 5).
(5) £ = a cos art and ?/ = 6 sin art.
(6) a; = a sin (art 5) and y = 6 cos (art + 5).
(7) x = a sin (art ~) and y = 6 sin (ort + -)
\ */ \ ^/
(8) £ = acos(coi + ) and y = 6sin (coi )-
\ «V \ «V
(9) ^ = a cos (ut j) and y = 6 cosf
(10) .-r = a cos (art + 50 and ?/ = 6 sin (co^ + 5o).
241. Physical Pendulum. Any rigid body which is free to oscillate under the action of its own weight is called a physical or a compound pendulum. Let A, Fig. 137, be a rigid body which is free to oscillate about a horizontal axis through the point 0 and perpendicular to the plane of the paper. Further let c denote the position of the center of mass and D its distance from the axis. Then the torque equation gives
/- = mgDsmd, (X)
where m is the mass of the body and 0 the angular displacement from the position of equilibrium.